# Properties

 Label 1950.2.z.e Level $1950$ Weight $2$ Character orbit 1950.z Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.z (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + 5 \zeta_{12}^{2} q^{11} -\zeta_{12}^{3} q^{12} + ( 4 \zeta_{12} - \zeta_{12}^{3} ) q^{13} -2 q^{14} -\zeta_{12}^{2} q^{16} + 2 \zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} + ( -2 + 2 \zeta_{12}^{2} ) q^{19} + 2 q^{21} -5 \zeta_{12} q^{22} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{2} q^{24} + ( -4 + \zeta_{12}^{2} ) q^{26} -\zeta_{12}^{3} q^{27} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + 5 \zeta_{12}^{2} q^{29} -11 q^{31} + \zeta_{12} q^{32} + 5 \zeta_{12} q^{33} -2 q^{34} -\zeta_{12}^{2} q^{36} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{37} -2 \zeta_{12}^{3} q^{38} + ( 4 - \zeta_{12}^{2} ) q^{39} + 2 \zeta_{12}^{2} q^{41} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{42} -11 \zeta_{12} q^{43} + 5 q^{44} + ( -1 + \zeta_{12}^{2} ) q^{46} + 9 \zeta_{12}^{3} q^{47} -\zeta_{12} q^{48} -3 \zeta_{12}^{2} q^{49} + 2 q^{51} + ( 3 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} -6 \zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} + 2 \zeta_{12}^{3} q^{57} -5 \zeta_{12} q^{58} + ( -15 + 15 \zeta_{12}^{2} ) q^{59} + ( -10 + 10 \zeta_{12}^{2} ) q^{61} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{62} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} - q^{64} -5 q^{66} + ( 16 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{67} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + ( 1 - \zeta_{12}^{2} ) q^{69} + \zeta_{12} q^{72} + 6 \zeta_{12}^{3} q^{73} + ( -3 + 3 \zeta_{12}^{2} ) q^{74} + 2 \zeta_{12}^{2} q^{76} + 10 \zeta_{12}^{3} q^{77} + ( -3 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{78} + 11 q^{79} -\zeta_{12}^{2} q^{81} -2 \zeta_{12} q^{82} -6 \zeta_{12}^{3} q^{83} + ( 2 - 2 \zeta_{12}^{2} ) q^{84} + 11 q^{86} + 5 \zeta_{12} q^{87} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{88} + 2 \zeta_{12}^{2} q^{89} + ( 2 + 6 \zeta_{12}^{2} ) q^{91} -\zeta_{12}^{3} q^{92} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{93} -9 \zeta_{12}^{2} q^{94} + q^{96} + 2 \zeta_{12} q^{97} + 3 \zeta_{12} q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{6} + 2q^{9} + 10q^{11} - 8q^{14} - 2q^{16} - 4q^{19} + 8q^{21} + 2q^{24} - 14q^{26} + 10q^{29} - 44q^{31} - 8q^{34} - 2q^{36} + 14q^{39} + 4q^{41} + 20q^{44} - 2q^{46} - 6q^{49} + 8q^{51} + 2q^{54} - 4q^{56} - 30q^{59} - 20q^{61} - 4q^{64} - 20q^{66} + 2q^{69} - 6q^{74} + 4q^{76} + 44q^{79} - 2q^{81} + 4q^{84} + 44q^{86} + 4q^{89} + 20q^{91} - 18q^{94} + 4q^{96} + 20q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 1.73205 1.00000i 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i −1.73205 + 1.00000i 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 1.73205 + 1.00000i 1.00000i 0.500000 0.866025i 0
1849.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i −1.73205 1.00000i 1.00000i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.z.e 4
5.b even 2 1 inner 1950.2.z.e 4
5.c odd 4 1 390.2.i.f 2
5.c odd 4 1 1950.2.i.d 2
13.c even 3 1 inner 1950.2.z.e 4
15.e even 4 1 1170.2.i.a 2
65.n even 6 1 inner 1950.2.z.e 4
65.o even 12 1 5070.2.b.h 2
65.q odd 12 1 390.2.i.f 2
65.q odd 12 1 1950.2.i.d 2
65.q odd 12 1 5070.2.a.d 1
65.r odd 12 1 5070.2.a.p 1
65.t even 12 1 5070.2.b.h 2
195.bl even 12 1 1170.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.f 2 5.c odd 4 1
390.2.i.f 2 65.q odd 12 1
1170.2.i.a 2 15.e even 4 1
1170.2.i.a 2 195.bl even 12 1
1950.2.i.d 2 5.c odd 4 1
1950.2.i.d 2 65.q odd 12 1
1950.2.z.e 4 1.a even 1 1 trivial
1950.2.z.e 4 5.b even 2 1 inner
1950.2.z.e 4 13.c even 3 1 inner
1950.2.z.e 4 65.n even 6 1 inner
5070.2.a.d 1 65.q odd 12 1
5070.2.a.p 1 65.r odd 12 1
5070.2.b.h 2 65.o even 12 1
5070.2.b.h 2 65.t even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{17}^{4} - 4 T_{17}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$( 25 - 5 T + T^{2} )^{2}$$
$13$ $$169 - 22 T^{2} + T^{4}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 4 + 2 T + T^{2} )^{2}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( 25 - 5 T + T^{2} )^{2}$$
$31$ $$( 11 + T )^{4}$$
$37$ $$81 - 9 T^{2} + T^{4}$$
$41$ $$( 4 - 2 T + T^{2} )^{2}$$
$43$ $$14641 - 121 T^{2} + T^{4}$$
$47$ $$( 81 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 225 + 15 T + T^{2} )^{2}$$
$61$ $$( 100 + 10 T + T^{2} )^{2}$$
$67$ $$65536 - 256 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$( -11 + T )^{4}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 4 - 2 T + T^{2} )^{2}$$
$97$ $$16 - 4 T^{2} + T^{4}$$